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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [215,4,Mod(1,215)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("215.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(215, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 215.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.6854106512\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 71 x^{11} + 359 x^{10} + 1706 x^{9} - 8844 x^{8} - 15698 x^{7} + 87775 x^{6} + \cdots - 150024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 5) q^{4} - 5 q^{5} + (\beta_{12} + \beta_{10} + \cdots + \beta_1) q^{6} + ( - \beta_{12} - \beta_{10} - \beta_{8} + \cdots + 2) q^{7} + (\beta_{5} + \beta_{4} + 7 \beta_1) q^{8}+ \cdots + ( - 41 \beta_{12} - 13 \beta_{11} + \cdots + 282) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 5 q^{2} + 5 q^{3} + 63 q^{4} - 65 q^{5} + 8 q^{6} + 34 q^{7} + 33 q^{8} + 152 q^{9} - 25 q^{10} + 29 q^{11} + 213 q^{12} + 221 q^{13} - 39 q^{14} - 25 q^{15} + 627 q^{16} + 122 q^{17} + 451 q^{18}+ \cdots + 3989 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 71 x^{11} + 359 x^{10} + 1706 x^{9} - 8844 x^{8} - 15698 x^{7} + 87775 x^{6} + \cdots - 150024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 183900819777 \nu^{12} + 1087641021182 \nu^{11} + 12651740645745 \nu^{10} + \cdots - 12\!\cdots\!44 ) / 560141389390944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 99147116741 \nu^{12} + 141310661389 \nu^{11} + 8655436280424 \nu^{10} + \cdots - 21\!\cdots\!32 ) / 140035347347736 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 99147116741 \nu^{12} - 141310661389 \nu^{11} - 8655436280424 \nu^{10} + \cdots + 21\!\cdots\!32 ) / 140035347347736 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2524634162195 \nu^{12} + 8508939606930 \nu^{11} + 195399771118707 \nu^{10} + \cdots - 30\!\cdots\!80 ) / 560141389390944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5122960274213 \nu^{12} - 18531109766690 \nu^{11} - 386208967728201 \nu^{10} + \cdots + 44\!\cdots\!04 ) / 560141389390944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2778616285439 \nu^{12} - 9139801748428 \nu^{11} - 215324776305465 \nu^{10} + \cdots + 32\!\cdots\!28 ) / 280070694695472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3300315870627 \nu^{12} - 11483891864678 \nu^{11} - 251831115806871 \nu^{10} + \cdots + 33\!\cdots\!00 ) / 280070694695472 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5348731445779 \nu^{12} - 18778403477162 \nu^{11} - 407584283505063 \nu^{10} + \cdots + 53\!\cdots\!88 ) / 280070694695472 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3553543192744 \nu^{12} - 12695394450377 \nu^{11} - 269530050237501 \nu^{10} + \cdots + 34\!\cdots\!68 ) / 140035347347736 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22069017164951 \nu^{12} + 77562841870766 \nu^{11} + \cdots - 22\!\cdots\!84 ) / 560141389390944 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{12} - 3 \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} + \cdots + 295 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{12} + 6 \beta_{11} + 2 \beta_{10} + 7 \beta_{9} + 5 \beta_{7} - 4 \beta_{6} + 44 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 57 \beta_{12} - 19 \beta_{11} - 124 \beta_{10} - 76 \beta_{9} + 90 \beta_{8} + 61 \beta_{7} + \cdots + 7489 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 416 \beta_{12} + 229 \beta_{11} + 119 \beta_{10} + 380 \beta_{9} - 75 \beta_{8} + 287 \beta_{7} + \cdots - 656 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2385 \beta_{12} - 1113 \beta_{11} - 4103 \beta_{10} - 3517 \beta_{9} + 2933 \beta_{8} + 2417 \beta_{7} + \cdots + 198777 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14833 \beta_{12} + 6347 \beta_{11} + 5291 \beta_{10} + 15323 \beta_{9} - 4783 \beta_{8} + 12159 \beta_{7} + \cdots - 49864 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 87359 \beta_{12} - 46415 \beta_{11} - 126869 \beta_{10} - 135673 \beta_{9} + 87055 \beta_{8} + \cdots + 5427199 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 487176 \beta_{12} + 156517 \beta_{11} + 205100 \beta_{10} + 551328 \beta_{9} - 210937 \beta_{8} + \cdots - 2579554 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2984888 \beta_{12} - 1693199 \beta_{11} - 3827031 \beta_{10} - 4783096 \beta_{9} + 2512477 \beta_{8} + \cdots + 151243286 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.54586
−4.58192
−3.85655
−1.60981
−1.55387
−0.574232
1.07999
1.48363
1.49110
3.37058
4.95406
5.00036
5.34253
−5.54586 7.22510 22.7566 −5.00000 −40.0694 19.6305 −81.8381 25.2021 27.7293
1.2 −4.58192 −5.65953 12.9940 −5.00000 25.9315 33.2121 −22.8820 5.03029 22.9096
1.3 −3.85655 −1.06624 6.87300 −5.00000 4.11199 −26.7599 4.34632 −25.8631 19.2828
1.4 −1.60981 5.15669 −5.40851 −5.00000 −8.30129 −27.8490 21.5852 −0.408562 8.04905
1.5 −1.55387 −3.84276 −5.58548 −5.00000 5.97117 −7.45782 21.1101 −12.2332 7.76937
1.6 −0.574232 7.63766 −7.67026 −5.00000 −4.38579 29.1073 8.99837 31.3339 2.87116
1.7 1.07999 −8.00576 −6.83362 −5.00000 −8.64615 −29.4375 −16.0202 37.0921 −5.39996
1.8 1.48363 −3.89647 −5.79884 −5.00000 −5.78092 4.63774 −20.4724 −11.8175 −7.41815
1.9 1.49110 −4.04607 −5.77661 −5.00000 −6.03310 18.5682 −20.5423 −10.6293 −7.45551
1.10 3.37058 7.84859 3.36079 −5.00000 26.4543 20.4108 −15.6368 34.6004 −16.8529
1.11 4.95406 9.21028 16.5427 −5.00000 45.6283 −21.8179 42.3212 57.8292 −24.7703
1.12 5.00036 −8.27614 17.0036 −5.00000 −41.3837 1.68664 45.0211 41.4945 −25.0018
1.13 5.34253 2.71465 20.5426 −5.00000 14.5031 20.0688 67.0095 −19.6307 −26.7127
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(43\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.4.a.c ✓ 13
3.b odd 2 1 1935.4.a.m 13
5.b even 2 1 1075.4.a.e 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.4.a.c ✓ 13 1.a even 1 1 trivial
1075.4.a.e 13 5.b even 2 1
1935.4.a.m 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 5 T_{2}^{12} - 71 T_{2}^{11} + 359 T_{2}^{10} + 1706 T_{2}^{9} - 8844 T_{2}^{8} + \cdots - 150024 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(215))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 5 T^{12} + \cdots - 150024 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots + 1352586512 \) Copy content Toggle raw display
$5$ \( (T + 5)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots + 40\!\cdots\!20 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots - 12\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots - 52\!\cdots\!40 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 35\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 91\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 38\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 61\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 87\!\cdots\!30 \) Copy content Toggle raw display
$43$ \( (T + 43)^{13} \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 15\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 14\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 66\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 17\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 20\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 21\!\cdots\!34 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 28\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
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